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Math Help - Convergence True or False Proof

  1. #1
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    Convergence True or False Proof

    If a_k \to 0 as k \to \infty, and \displaystyle\sum_{k=1}^{\infty} a_k converges, then a_k \downarrow 0 as k \to \infty.

    I think this is false and my example is (-1)^k \frac{1}{k^2}. What you think?
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  2. #2
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    Quote Originally Posted by ThreeLeftsAndHome View Post
    If a_k \to 0 as k \to \infty, and \displaystyle\sum_{k=1}^{\infty} a_k converges, then a_k \downarrow 0 as k \to \infty.
    first information is irrelevant since given the convergence of the series, then a_k\to0 as k\to\infty.

    oh, i think is misread the problem, wonder what does mean a_k\downarrow0.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    first information is irrelevant since given the convergence of the series, then a_k\to0 as k\to\infty.

    oh, i think is misread the problem, wonder what does mean a_k\downarrow0.
    I think it means that a_k is monotone decreasing.
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  4. #4
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    ah well, but it doesn't say if a_n\ge0, so the counterexample is correct.
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